Solomon Feferman writes:
> > I am intrigued by your thesis that "monsters" ...
> > are best viewed simply as counterexamples showing that certain
> > hypotheses cannot be omitted from certain theorems ....
>
> This *is* what I say about the older pathologies (nowhere
> differentiable continuous function, space-filling curve, etc.), and
> I duly consider whether it is also applicable to the B-T theorem,
> but my conclusion p. 11 is that there is no obvious way to do so.
Sol, I read that conclusion, but I guess I didn't really take it in,
because I didn't understand it, because I have always viewed
Banach/Tarski in what I thought was the obvious way, as a stark
counterexample in the same vein as the older pathologies. Namely, B-T
is a counterexample showing that the hypothesis of measurability
cannot be omitted in the following intuitive theorem: If two bounded
solids are equivalent by decomposition into finitely many measurable
pieces, then they have the same volume. Or, replace measurable by
some other "niceness" property, such as polyhedral, or Borel. But
apparently you have a critique of this viewpoint, on the grounds that
the notion of equivalence by decomposition used here is inappropriate,
because it is views the solids as mere sets of points.
Anyway, I won't try to pursue this further here on the FOM list,
because unfortunately you are no longer a subscriber.
Best regards,
-- Steve